Algebraic hyperbolicity of very general hypersurfaces in products of projective spaces

Abstract: A complex algebraic variety is said to be hyperbolic if it contains no entire curves, which are non-constant holomorphic images of the complex line. Demailly introduced algebraic hyperbolicity as an algebraic version of this property, and it has since been well-studied as a means for understanding Kobayashi’s conjecture, which says that a generic hypersurface in dimensional projective space is hyperbolic whenever its degree is large enough. In this talk, we study the algebraic hyperbolicity of very general hypersurfaces of high bi-degrees in Pm x Pn and completely classify them by their bi-degrees, except for a few cases in P3 x P1. We present three techniques to do that, which build on past work by Ein, Voisin, Pacienza, Coskun and Riedl, and others. As another application of these techniques, we simplify a proof of Voisin (1988) of the algebraic hyperbolicity of generic high-degree projective hypersurfaces.

algebraic geometry

Audience: researchers in the topic

Algebraic Geometry NorthEastern Series (AGNES)